Minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a
Definitions
Minimal surfaces can be defined in several equivalent ways in . The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.[1]
- Local least area definition: A surface is minimal if and only if every point p ∈ M has a neighbourhood, bounded by a simple closed curve, which has the least area among all surfaces having the same boundary.
This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. This property establishes a connection with soap films; a soap film deformed to have a wire frame as boundary will minimize area.
- Variational definition: A surface is minimal if and only if it is a critical point of the area functional for all compactly supported variations.
This definition makes minimal surfaces a 2-dimensional analogue to
- Mean curvature definition: A surface is minimal if and only if its mean curvature is equal to zero at all points.
A direct implication of this definition is that every point on the surface is a
- Differential equation definition: A surface is minimal if and only if it can be locally expressed as the graph of a solution of
The partial differential equation in this definition was originally found in 1762 by
- Energy definition: A conformal immersion is minimal if and only if it is a critical point of the Dirichlet energy for all compactly supported variations, or equivalently if any point has a neighbourhood with least energy relative to its boundary.
This definition ties minimal surfaces to
- Harmonic definition: If is an isometric immersion of a Riemann surface into 3-space, then is said to be minimal whenever is a harmonic function on for each .
A direct implication of this definition and the
minimal surfaces in .- Gauss map definition: A surface is minimal if and only if its stereographically projected Gauss map is meromorphic with respect to the underlying Riemann surfacestructure, and is not a piece of a sphere.
This definition uses that the mean curvature is half of the
The local least area and variational definitions allow extending minimal surfaces to other
History
Minimal surface theory originates with
He did not succeed in finding any solution beyond the plane. In 1776
By expanding Lagrange's equation to
Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces. While these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface.
Progress had been fairly slow until the middle of the century when the
Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. The complete solution of the Plateau problem by Jesse Douglas and Tibor Radó was a major milestone. Bernstein's problem and Robert Osserman's work on complete minimal surfaces of finite total curvature were also important.
Another revival began in the 1980s. One cause was the discovery in 1982 by Celso Costa of a surface that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in of finite topological type. This not only stimulated new work on using the old parametric methods, but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem" (when using the conjugate surface method to determine surface patches that can be assembled into a larger symmetric surface, certain parameters need to be numerically matched to produce an embedded surface). Another cause was the verification by H. Karcher that the triply periodic minimal surfaces originally described empirically by Alan Schoen in 1970 actually exist. This has led to a rich menagerie of surface families and methods of deriving new surfaces from old, for example by adding handles or distorting them.
Currently the theory of minimal surfaces has diversified to minimal submanifolds in other ambient geometries, becoming relevant to mathematical physics (e.g. the
Examples
Classical examples of minimal surfaces include:
- the trivialcase
- catenoids: minimal surfaces made by rotating a catenary once around its directrix
- helicoids: A surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity
Surfaces from the 19th century golden age include:
- Schwarz minimal surfaces: triply periodic surfaces that fill
- Riemann's minimal surface: A posthumously described periodic surface
- the Enneper surface
- the Henneberg surface: the first non-orientable minimal surface
- Bour's minimal surface
- the Neovius surface: a triply periodic surface
Modern surfaces include:
- the Gyroid: One of Schoen's surfaces from 1970, a triply periodic surface of particular interest for liquid crystal structure
- the Saddle tower family: generalisations of Scherk's second surface
- Costa's minimal surface: Famous conjecture disproof. Described in 1982 by Celso Costa and later visualized by Jim Hoffman. Jim Hoffman, David Hoffman and William Meeks III then extended the definition to produce a family of surfaces with different rotational symmetries.
- the Chen–Gackstatter surface family, adding handles to the Enneper surface.
Generalisations and links to other fields
Minimal surfaces can be defined in other
The definition of minimal surfaces can be generalized/extended to cover constant-mean-curvature surfaces: surfaces with a constant mean curvature, which need not equal zero.
The curvature lines of an isothermal surface form an isothermal net.[5]
In discrete differential geometry discrete minimal surfaces are studied: simplicial complexes of triangles that minimize their area under small perturbations of their vertex positions.[6] Such discretizations are often used to approximate minimal surfaces numerically, even if no closed form expressions are known.
Brownian motion on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces.[7]
Minimal surfaces have become an area of intense scientific study, especially in the areas of molecular engineering and materials science, due to their anticipated applications in self-assembly of complex materials.[8] The endoplasmic reticulum, an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface.[9]
In the fields of
Structures with minimal surfaces can be used as tents.
Minimal surfaces are part of the
In the art world, minimal surfaces have been extensively explored in the sculpture of Robert Engman (1927–2018), Robert Longhurst (1949– ), and Charles O. Perry (1929–2011), among others.
See also
- Bernstein's problem
- Bilinear interpolation
- Bryant surface
- Curvature
- Enneper–Weierstrass parameterization
- Harmonic map
- Harmonic morphism
- Plateau's problem
- Schwarz minimal surface
- Soap bubble
- Surface Evolver
- Stretched grid method
- Tensile structure
- Triply periodic minimal surface
- Weaire–Phelan structure
References
- MR 2801776.
- ^ J. L. Lagrange. Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies. Miscellanea Taurinensia 2, 325(1):173{199, 1760.
- ^ J. B. Meusnier. Mémoire sur la courbure des surfaces. Mém. Mathém. Phys. Acad. Sci. Paris, prés. par div. Savans, 10:477–510, 1785. Presented in 1776.
- ^ See (Nishikawa 2002) about variational definition.
- ^ "Isothermal surface - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2022-09-04.
- MR 1246481.
- S2CID 15228691.
- S2CID 3928702.
- PMID 23870120.
- ISBN 978-0-19-923072-3(page 417)
- ^ "AD Classics: Olympiastadion (Munich Olympic Stadium) / Behnisch and Partners & Frei Otto". ArchDaily. 2011-02-11. Retrieved 2022-09-04.
- ^ "Expo 67 German Pavilion". Architectuul. Retrieved 2022-09-04.
Further reading
Textbooks
- R. Courant. Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces. Appendix by M. Schiffer. Interscience Publishers, Inc., New York, N.Y., 1950. xiii+330 pp.
- ISBN 0-914098-18-7
- MR0852409
- Johannes C.C. Nitsche. Lectures on minimal surfaces. Vol. 1. Introduction, fundamentals, geometry and basic boundary value problems. Translated from the German by Jerry M. Feinberg. With a German foreword. Cambridge University Press, Cambridge, 1989. xxvi+563 pp. ISBN 0-521-24427-7
- Nishikawa, Seiki (2002). Variational problems in geometry. Translations of mathematical monographs; Iwanami series in modern mathematics. Vol. 205. Translated by Abe, Kinetsu. Providence, R. I. : )
- 西川青季 (1998). 幾何学的変分問題. 岩波講座現代数学の基礎 (in Japanese). Vol. 28. Tokyo: ISBN 4-00-010642-2.
- 西川青季 (1998). 幾何学的変分問題. 岩波講座現代数学の基礎 (in Japanese). Vol. 28. Tokyo:
- Ulrich Dierkes, Stefan Hildebrandt, and Friedrich Sauvigny. Minimal surfaces. Revised and enlarged second edition. With assistance and contributions by A. Küster and R. Jakob. Grundlehren der Mathematischen Wissenschaften, 339. Springer, Heidelberg, 2010. xvi+688 pp. MR2566897
- ISBN 978-0-8218-5323-8
Online resources
- Karcher, Hermann; Polthier, Konrad (1995). "Touching Soap Films - An introduction to minimal surfaces". Retrieved December 27, 2006. (graphical introduction to minimal surfaces and soap films.)
- Jacek Klinowski. "Periodic Minimal Surfaces Gallery". Retrieved February 2, 2009. (A collection of minimal surfaces with classical and modern examples)
- Martin Steffens and Christian Teitzel. "Grape Minimal Surface Library". Retrieved October 27, 2008. (A collection of minimal surfaces)
- Various (2000). "EG-Models". Retrieved September 28, 2004. (Online journal with several published models of minimal surfaces)